Displacement

Figure 1. Motion of a continuum body.

A change in the configuration of a continuum body results in a
displacement. The
displacement of a body has two components: a rigid-body
displacement and a deformation. A rigid-body displacement consist
of a simultaneous translation and rotation of the body without
changing its shape or size. Deformation implies the change in shape
and/or size of the body from an initial or undeformed configuration
to a current or deformed configuration
(Figure 1).

If after a displacement of the continuum there is a relative
displacement between particles, a deformation has occurred. On the
other hand, if after displacement of the continuum the relative
displacement between particles in the current configuration is zero
i.e. the distance between particles remains unchanged, then there
is no deformation and a rigid-body displacement is said to have
occurred.

The vector joining the positions of a particle
in the undeformed configuration and deformed configuration is
called the displacement vector,
in the Lagrangian description, or ,
in the Eulerian description.

A displacement field is a vector field of all
displacement vectors for all particles in the body, which relates
the deformed configuration with the undeformed configuration. It is
convenient to do the analysis of deformation or motion of a
continuum body in terms of the displacement field. In general, the
displacement field is expressed in terms of the material
coordinates as

or in terms of the spatial coordinates as

where
are the direction cosines between the material and spatial
coordinate systems with unit vectors
and ,
respectively. Thus

and the relationship between
and
is then given by

Knowing that

then

It is common to superimpose the coordinate systems for the
undeformed and deformed configurations, which results in ,
and the direction cosines become Kronecker deltas, i.e.

Thus, we have

or in terms of the spatial coordinates as

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Displacement gradient
tensor

The partial differentiation of the displacement vector with
respect to the material coordinates yields the material
displacement gradient tensor.
Thus we have,

where
is the deformation gradient tensor.

Similarly, the partial differentiation of the displacement
vector with respect to the spatial coordinates yields the
spatial displacement gradient tensor.
Thus we have,

Deformation gradient
tensor

Figure 2. Deformation of a continuum body.

Consider a particle or material point
with position vector
in the undeformed configuration (Figure 2). After a displacement of
the body, the new position of the particle indicated by
in the new configuration is given by the vector position .
The coordinate systems for the undeformed and deformed
configuration can be superimposed for convenience.

Consider now a material point
neighboring ,
with position vector .
In the deformed configuration this particle has a new position
given by the position vector .
Assuming that the line segments
and
joining the particles
and
in both the undeformed and deformed configuration, respectively, to
be very small, then we can expressed them as
and .
Thus from Figure 2 we have

where
is the relative displacement vector, which
represents the relative displacement of
with respect to
in the deformed configuration.

For an infinitesimal element ,
and assuming continuity on the displacement field, it is possible
to use a Taylor series
expansion around point ,
neglecting higher-order terms, to approximate the components of the
relative displacement vector for the neighboring particle
as

Thus, the previous equation
can be written as

The material deformation gradient tensor
is a second-order tensor that
represents the gradient of the mapping function or functional
relation ,
which describes the motion of a continuum. The material
deformation gradient tensor characterizes the local deformation at
a material point with position vector ,
i.e. deformation at neighbouring points, by transforming (linear transformation) a material line
element emanating from that point from the reference configuration
to the current or deformed configuration, assuming continuity in
the mapping function ,
i.e differentiable
function of
and time ,
which implies that cracks
and voids do not open or close during the deformation. Thus we
have,

The deformation gradient tensor
is related to both the reference and current configuration, as seen
by the unit vectors
and ,
therefore it is a two-point tensor.

Transformation
of a surface and volume element

To transform quantities that are defined with respect to areas
in a deformed configuration to those relative to areas in a
reference configuration, and vice versa, we use the Nanson's
relation, expressed as

where
is an area of a region in the deformed configuration,
is the same area in the reference configuration, and
is the outward normal to the area element in the current
configuration while
is the outward normal in the reference configuration,
is the deformation gradient, and .

Derivation of Nanson's relation

To see how this formula is derived, we start with the oriented
area elements

in the reference and current configurations:

The reference and current volumes of an element are

where .

Therefore,

or,

or,

So we get

or,

Polar
decomposition of the deformation gradient tensor

Figure 3. Representation of the polar decomposition of the
deformation gradient

The deformation gradient ,
like any second-order tensor, can be decomposed, using the polar
decomposition theorem, into a product of two second-order
tensors (Truesdell and Noll, 1965): an orthogonal tensor and a
positive definite symmetric tensor, i.e.

where the tensor
is a proper orthogonal tensor, i.e.
and ,
representing a rotation; the tensor
is the right stretch tensor; and
the left stretch tensor. The terms right and
left means that they are to the right and left of the
rotation tensor ,
respectively.
and
are both positive definite, i.e.
and ,
and symmetric
tensors, i.e.
and ,
of second order.

This decomposition implies that the deformation of a line
element
in the undeformed configuration onto
in the deformed configuration, i.e. ,
may be obtained either by first stretching the element by ,
i.e. ,
followed by a rotation ,
i.e. ;
or equivalently, by applying a rigid rotation
first, i.e. ,
followed later by a stretching ,
i.e.
(See Figure 3).

It can be shown that,

so that
and
have the same eigenvalues or principal stretches, but
different eigenvectors or principal
directions
and ,
respectively. The principal directions are related by

This polar decomposition is unique as
is non-symmetric.

Deformation
tensors

Several rotation-independent deformation tensors are used in
mechanics. In solid mechanics, the most popular of these are the
right and left Cauchy-Green deformation tensors. The Finger
deformation tensor is mainly used in describing the motion of
nonlinear fluids.

Since a pure rotation should not induce any stresses in a
deformable body, it is often convenient to use rotation-independent
measures of deformation in continuum mechanics. As a rotation
followed by its inverse rotation leads to no change ()
we can exclude the rotation by multiplying
by its transpose.

The Right
Cauchy-Green deformation tensor

In 1839, George
Green introduced a deformation tensor known as the right
Cauchy-Green deformation tensor or Green's deformation
tensor, defined as:

Physically, the Cauchy-Green tensor gives us the square of local
change in distances due to deformation, i.e.

For nearly incompressible materials, a slightly different set of
invariants is used:

The Cauchy or Finger
deformation tensor

Earlier in 1828 [1], Augustin Louis Cauchy introduced a
deformation tensor defined as the inverse of the left Cauchy-Green
deformation tensor, ,
which is often called the Cauchy deformation tensor or
Finger deformation tensor, named after Josef Finger
(1894).

Spectral
representation

Therefore the uniqueness of the spectral decomposition also
implies that .
The left stretch ()
is also called the spatial stretch tensor while the right
stretch ()
is called the material stretch tensor.

The effect of
acting on
is to stretch the vector by
and to rotate it to the new orientation ,
i.e,

In a similar vein,

Examples

Uniaxial extension of an incompressible
material

This is the case where a specimen is stretched in 1-direction
with a stretch ratio of .
If the volume remains constant, the contraction in the other two
directions is such that
or .
Then:

Simple shear

Rigid body rotation

Derivatives of stretch

Derivatives of
the stretch with respect to the right Cauchy-Green deformation
tensor are used to derive the stress-strain relations of many
solids, particularly hyperelastic materials. These
derivatives are

and follow from the observations that

Finite
strain tensors

The concept of strain is used to evaluate how much a
given displacement differs locally from a rigid body displacement
(Ref. Lubliner). One of such strains for large deformations is the
Lagrangian finite strain tensor, also called the
Green-Lagrangian strain tensor or Green - St-Venant
strain tensor, defined as

or as a function of the displacement gradient tensor

or

The Green-Lagrangian strain tensor is a measure of how much
differs from .
It can be shown that this tensor is a special case of a general
formula for Lagrangian strain tensors (Hill 1968):

For different values of
we have:

The Eulerian-Almansi finite strain tensor, referenced
to the deformed configuration, i.e. Eulerian description, is
defined as

or as a function of the displacement gradients we have

Derivation of the Lagrangian and Eulerain finite strain
tensors

A measure of deformation is the difference between the squares
of the differential line element ,
in the undeformed configuration, and ,
in the deformed configuration (Figure 2). Deformation has occurred
if the difference is non zero, otherwise a rigid-body displacement
has occurred. Thus we have,

In the Lagrangian description, using the material coordinates as
the frame of reference, the linear transformation between the
differential lines is

Then we have,

where
are the components of the right Cauchy-Green deformation
tensor, .
Then, replacing this equation into the first equation we have,

or

where ,
are the components of a second-order tensor called the Green -
St-Venant strain tensor or the Lagrangian finite strain
tensor,

In the Eulerian description, using the spatial coordinates as
the frame of reference, the linear transformation between the
differential lines is

where
are the components of the spatial deformation gradient
tensor, .
Thus we have

where the second order tensor
is called Cauchy's deformation tensor, .
Then we have,

or

where ,
are the components of a second-order tensor called the
Eulerian-Almansi finite strain tensor,

Both the Lagrangian and Eulerian finite strain tensors can be
conveniently expressed in terms of the displacement gradient
tensor. For the Lagrangian strain tensor, first we
differentiate the displacement vector
with respect to the material coordinates
to obtain the material displacement gradient tensor,

Replacing this equation into the expression for the Lagrangian
finite strain tensor we have

or

Similarly, the Eulerian-Almansi finite strain tensor can be
expressed as

Stretch
ratio

The stretch ratio is a measure of the
extensional or normal strain of a differential line element, which
can be defined at either the undeformed configuration or the
deformed configuration.

The stretch ratio for the differential element
(Figure) in the direction of the unit vector
at the material point ,
in the undeformed configuration, is defined as

where
is the deformed magnitude of the differential element .

Similarly, the stretch ratio for the differential element
(Figure), in the direction of the unit vector
at the material point ,
in the deformed configuration, is defined as

The normal strain
in any direction
can be expressed as a function of the stretch ratio,

This equation implies that the normal strain is zero, i.e. no
deformation, when the stretch is equal to unity. Some materials,
such as elastometers can sustain stretch ratios of 3 or 4 before
they fail, whereas traditional engineering materials, such as
concrete or steel, fail at much lower stretch ratios, perhaps of
the order of 1.001 (reference?)

Physical
interpretation of the finite strain tensor

The diagonal components
of the Lagrangian finite strain tensor are related to the normal
strain, e.g.

where
is the normal strain or engineering strain in the direction .

The off-diagonal components
of the Lagrangian finite strain tensor are related to shear strain,
e.g.

where
is the change in the angle between two line elements that were
originally perpendicular with directions
and ,
respectively.

Under certain circumstances, i.e. small displacements and small
displacement rates, the components of the Lagrangian finite strain
tensor may be approximated by the components of the infinitesimal strain
tensor

Derivation of the physical interpretation of the Lagrangian and
Eulerian finite strain tensors

The stretch ratio for the differential element
(Figure) in the direction of the unit vector
at the material point ,
in the undeformed configuration, is defined as

where
is the deformed magnitude of the differential element .

Similarly, the stretch ratio for the differential element
(Figure), in the direction of the unit vector
at the material point ,
in the deformed configuration, is defined as

The square of the stretch ratio is defined as

Knowing that

we have

where
and
are unit vectors.

The normal strain or engineering strain
in any direction
can be expressed as a function of the stretch ratio,

Thus, the normal strain in the direction
at the material point
may be expressed in terms of the stretch ratio as

solving for
we have

The shear strain, or change in angle between two line
elements
and
initially perpendicular, and oriented in the principal directions
and ,
respectivelly, can also be expressed as a function of the stretch
ratio. From the dot
product between the deformed lines
and
we have

where
is the angle between the lines
and
in the deformed configuration. Defining
as the shear strain or reduction in the angle between two line
elements that were originally perpendicular, we have